Elektriseadmete raalprojekteerimine kursuse
metoodiline juhend: transformaator
Metoodiline juhend üliõpilastele ja juhendajatele
Studying guide in course on Numerical modelling and
design of electrical devices: transformers
Guide for students and supervisors
Project 1.0101-0278
Application of interdisciplinary and international team
and project based learning in Master Studies
IN 557
Studying guide
Objective
A construction and some design guidelines of the low-voltage low-power transformers are
described in this paper. The guidelines allow student to proceed through the design process,
modelling and analysis of the small-size transformers for adapted power supplies. As a result a
technical specification can be composed in order to build a prototype model.
This material could be efficiently used when studying or teaching transformers and could be a
complementary material for some classical textbooks describing transformer construction and
design. The purpose of this material is to achieve a complete design experience, where much
learning takes place, so that the student gets a good “hands-on” experience of exhaustively
analysing, designing and finally prototyping the device. The material avoids establishing a design
model that could be directly run in computer software in order to facilitate theoretical part of the
work. The material rather tries to give some calculation examples and analysis that could be used in
the design model that is actually engineered by a student. Guiding questions “Q” should encourage
the development of a creative thinking and an intelligence of analysis.
This design guide for students and supervisors are used during individual preparation period and it
is continued in the team-work session to study the design of transformers focusing on energy
conversion efficiency, cost effectiveness and cost analysis.
Content
1
2
3
Introduction ..............................................................................................................................................1
1.1
Design goals ........................................................................................................................................1
1.2
Design constrains ...............................................................................................................................2
1.3
Design optimization...........................................................................................................................2
Construction of a transformer ...............................................................................................................3
2.1
Transformers with a laminated core................................................................................................4
2.2
Winding configuration.......................................................................................................................6
Design of a transformer ..........................................................................................................................7
3.1
Functionality specification ................................................................................................................7
3.2
Geometric modelling .........................................................................................................................8
3.3
Size equations....................................................................................................................................12
3.4
Equivalent circuit of a transformer................................................................................................15
4
5
6
Numerical field modelling of a transformer ......................................................................................21
4.1
Model parameterization...................................................................................................................21
4.2
Magnetization....................................................................................................................................24
4.3
Loaded transformer..........................................................................................................................30
4.4
Temperature rise...............................................................................................................................30
4.5
Computation sequence ....................................................................................................................34
A three-phase transformer....................................................................................................................35
5.1
Construction......................................................................................................................................35
5.2
Electric circuits..................................................................................................................................36
Experimental work.................................................................................................................................37
6.1
Prototyping........................................................................................................................................37
6.2
Measurements ...................................................................................................................................37
References.........................................................................................................................................................39
1 Introduction
A learning process that bases on the individual efforts and develops the theoretical understanding
and the practical experience has undoubtedly a great input to the education of electrical engineers.
This particular course bases on the numeric modelling, design and prototyping of electrical devices
– in particular small power supply transformers. The objective in this course is to achieve a complete
design experience including the design, the actual construction and the testing of an electromagnetic
device such as a transformer [18] [24][25]. It is important to utilize the advantages of computer
aided design and manufacturing (CAD/ CAM) in order to make a device that meets all the
performance requirements of the device at minimum cost. A good design is an optimal utilization
of geometry, properties of media and performance requirements. Note that the design goal is to
exceed, not to just meet, the performance and reliability requirements. This comes from the
philosophy that given fixed resources the engineer's task is to get the most from these resources.
The design guidelines are focused on low and fixed frequency domain. In the longer perspective the
guideline material should be extended in order to design or to evaluate any kind of electromagnetic
energy converter – a transformer that operation mode is
•
Conventional
•
Switching
•
Resonance
and the common purpose is
•
Power transfer
•
Power conditioning
•
Galvanic separation
As a matter of fact the guideline material in the present size focuses to the low frequency
continuous mode single phase transformer in order to establish a good physical understanding,
which is essential in the computer aided design process [5][6][8][9][12][13][15][16][20][21][23].
Therefore it could be frustrating that the problems in electromagnetism are handled almost like
static by neglecting the influence of the time dependent fields. Anyway the switched mode power
supply (SMPS) will be a continuation to the present material [22].
1.1 Design goals
The performance requirement for the power transformer or the transformer functionality is obtained by
the electrical design factors, which are related to the supply and to the load such as power, voltage,
frequency and number of phases and the voltage regulation. The device of a certain power
supposes to fulfil certain requirements to insulation in order to withstand the temperature and
avoid flashovers. Thermal requirements and cooling conditions are stated in the specification.
Additionally the mechanical or electromagnetic conditions such as audible noise or shielding,
respectively, can be specified. Mechanical strength requirement, vibration, resonant modes, etc are
usually not prioritized as a design goal.
2
1. Introduction
1.2 Design constrains
The transformer principle of operation bases on Faraday’s law of induction. The challenge of
designing a transformer is to solve a coupled problem of multiphysics. This incorporates
electromagnetism, mechanics, heat transfer and the coupling between the different fields that is all
in accordance with energy conversion principle. Theoretical design factors are used to quantify the
physical parameters of the energy conversion process [19]. They can be broken into five areas:
electrical, dielectric, magnetic, mechanical and thermal. These can be also seen in turn as different
circuits with flows and flow densities, which are coupled into geometry. The flow densities such as
the electrical current, the magnetic flux and the loss power density can be considering as the
loadings that establish the relations between the size, the material properties and the performance
of the energy converter. In the case of a transformer one of the design constrains is to model and
specify properly its behaviour in the circuit application (including the effects of multiphysics).
1.3 Design optimization
In order to optimize a transformer it is necessary to find an appropriate geometric distribution of
materials and sources that fulfil the performance requirements – a device model. The optimal design
of an electric device is to choose a set of dimensions for the device (design variables), so that the
designed device will satisfy the design criteria (constraints) and minimize (or maximize) some
additional criterion (objective function). The objective functions, which could be chosen such as
minimal losses, minimal cost (and temperature) [3], minimal weight etc, are optimized [4][7].
2 Construction of a transformer
A transformer has at least a pair of coils coupled magnetically together. As a consequence some of the
magnetic flux produced by current in one of the coils links the turns of the other(s), and vice versa.
Magnetic flux and electric current ‘indicate’ the flow and the presence of magnetic and electrical
circuit(s). The fundamental purpose of any magnetic core or a circuit is to provide an easy path for
flux in order to facilitate flux linkage or magnetic coupling between two or more magnetic
elements. In an electromagnetic energy converter the magnetic elements are primary and the
secondary windings. The electromagnetic coupling between the magnetic and electric circuits obeys
Ampere’s and Faraday’s law (Figure 2.1).
φ
i
E
H
emf = ∫ E ⋅ dl = −
dφ
dt
mmf = ∫ H ⋅ dl = i
Figure 2.1 Ampere’s and Faraday’s law describing the static and the dynamic connection between the
electrical and the magnetical quantities. The reason for an electrical current is the
electromotive force (emf) such as for the magnetic flux is the magnetomotive force (mmf).
Transformer can be seen as an electromagnetic inductor that is loaded with a secondary winding.
As a matter of fact a transformer that instantaneously converts energy between the primary and the
secondary winding have to have high magnetic permeability of the magnetic circuit as the energy
storage in the transformer core is undesired parasitic effect. On contrary, a transformer operating in
switching mode needs to have intermediate energy storage that means a low net magnetic
permeability of the circuit.
Figure 2.2 Electromagnetic circuit with the shortest ‘contour’ length. The origin of shell and core type.
One possible, a geometrically ideal coupling of electric and magnetic circuit is shown in Figure 2.2.
It would not be complicated to imagine that ‘circuit A’ could be the electric circuit and ‘circuit B’ is
the magnetic circuit, or vice versa. From functionality point of view the electric circuit may consists
as many separate windings as needed. The windings are wound close to each other in order to
2. Construction of a transformer
4
reduce leakage flux and to reduce a voltage regulation. From manufacturability point of view, a
winding is composed from a number of electrically insulated wire strands so that the necessary
ampere turns or voltage per turn can be achieved. Similarly, the magnetic circuit could be made
from a magnetic wire. For the medium and high power, low frequency transformer it is important
that the magnetic core has low permeability, low magnetization losses and the core could be easily
assembled. This condition is fulfilled by using a stack of laminated electromagnetic steel sheets.
2.1 Transformers with a laminated core
Two general configurations of iron and copper in a transformer are shell or core type of transformer.
Figure 2.3 shows the cross-section of a transformer that is in magnetic flux plane and perpendicular
to the electric current. Figure 2.4 shows on contrary the cross-section of a transformer that is in
electric current plane and perpendicular to the magnetic flux. Further investigation of the different
core types is described in chapter Error! Reference source not found.. Most small power
transformers are shell-type, in which the iron surrounds the copper. The magnetic circuit is divided
into two return paths on opposite sides of the core, as is done with the E and I laminations, as
shown. Another configuration, the core type, in which the copper surrounds the iron, is common
with larger high-voltage transformers, as well as with small ferrite-core toroidal transformers. Note
that both primary and secondary windings are placed on each leg. The laminations are L-shaped to
make a core of uniform cross-section with a central rectangular window. Core transformers are
expectedly easier to insulate and to cool. If the primary winding were placed on one leg and the
secondary on the other, as in the diagrams of transformers in texts, leakage flux would be excessive,
and the transformer would have poor regulation (variation of voltage with load). This kind of
transformer analysis is described in chapter 4. There are other special types of construction such as
distributed core, etc, but all can be classified as shell or core.
The important feature of the transformer core, which is made as a stack of laminations, is a high
permeability. This can be achieved, especially in grain oriented electromagnetic steels, where the
materials have a high permeability in the rolling direction and not as good any other direction.
Grain-oriented steel is inherently suitable for wound-core construction (right column in Figure 2.3
and Figure 2.4). The simplest way to build a wound core is to wind a continuous strip of steel on a
rectangular form. When the desired thickness has been built up the core is split into two halves for
a single magnetic flux loop. If the planar laminations are the same or the constant width of the steel
strip in the wound core, it is obvious that the magnetic circuit will have a rectangular cross-section.
However, if the narrower laminations are used for the outer layers, then it is possible to get moreor-less round cross-section of the magnetic core.
Another important feature of the laminated transformer core is the magnetization losses. If the
lamination thickness is reduced by factor k, then the eddy current loss is reduced by a factor k2. The
eddy current losses are proportional to electric resistivity of the material. Hysteresis losses are
usually independent of material impurity, which results at the same time higher resistivity and
hysteresis losses. The fringing fields in the lamination joins and stray flux in the perpendicular
direction of the lamination plate are significant source of losses and the local heating.
Q.1
What kind of shape should have discrete laminations in order to form a core and which
cases the advantage can be taken of the grain oriented steel?
Q.2
What kind of shape and proportions suppose to have a scrapless lamination (the
laminations stamp from a rectangle of electromagnetic steel sheet has no waste left
over)?
Q.3
How should look like the corresponding shell and core type transformer (2D/3D) for a
three-phase transformer?
2. Construction of a transformerConstruction of a transformer
Figure 2.3 The cross-section of a transformer on the plane of magnetic flux flow. The upper row
indicates a core-type transformer and the lower row a shell-type of transformer. The
transformers with planar laminations are shown in the left column and the wound-core
transformers in the right column. If the same electric conductor cross-section is considered then
the core-type of transformer are expectedly longer compared to shell-type of transformers, which
is due to additional insulation material.
Figure 2.4 The cross-section of a transformer on the plane of electric current flow. The upper row
indicates a core-type transformer and the lower row a shell-type of transformer. The
transformers with planar laminations are shown in the left column and the wound-core
transformers in the right column. The thinner cross-section of the winding in the core-type of
transformers gives expectedly the advantage of the lower thermal stress.
5
2. Construction of a transformer
6
2.2 Winding configuration
The usual winding arrangement is that the low voltage coil has been wound closest to the core and
the high-voltage coil has been wound on top of the low voltage coil, which is different that is
shown in Figure 2.3 and Figure 2.4. A sandwiched configuration where the low voltage section is
divided into two halves and the high voltage winding is placed between gives improved voltage
regulation. Windings can be wound also in sections in side to side as a radial discs instead of axial
layers. As a matter of fact the concentric coils reduce inefficient (leakage) area and thus improve the
voltage regulation. One way or another there normally will be a considerable thickness of insulation
between the primary and the secondary coil. The important in the winding configuration is the
capability to dissipate heat and to bear electrodynamical forces in the transient and short-circuit
conditions, which tends to round out of the flat sides of the loop coil. In practice, most of the
transformer coils are wound on a form, dipped in insulating varnish, backed in rigid mass, and then
are assembled with core. The good thermal contact between the core and the winding is important
in order to facilitate the heat dissipation.
Q.4
Why does the cooling problem become progressively more difficult as transformer size
increases?
Q.5
How should be the windings placed in order to take advantage of good backing and
improved heat dissipation, lower leakage and better voltage regulation?
3 Design of a transformer
In order to design a transformer, or to examine in more detail how it departs from idealistic, it is
necessary to understand how a transformer works, not just how to express its terminal relations in
an approximate way. It is also important to know how the properties of the iron core affect the
performance of the transformer. A real transformer becomes hot because of losses, and the output
voltage may vary with load even when the primary voltage is held constant. It is important not
simply to design a transformer that will do, but one that is economical, efficient and makes the best
use of available materials.
Transformer design consists of five parts: physical understanding, mathematical modelling, synthesis,
analysis and a cost functional [24]. Analysis generally means the action of taking something apart in
order to study it. Synthesis is an opposite process which results in a new creation. Design as a
process is originating and developing a plan for a new object. Design as a result is the final plan or
proposal to produce the new object. Designing normally requires considering functional, and many
other aspects of an object, which usually requires considerable research, thought, modelling,
iterative adjustment, and re-design. Generally, functional refers to something able to fulfil its
purpose or function. Design as a process can take many forms depending on the object being
designed and the individual or individuals participating. Optimize the magnet device design.
Optimized for a light-weight construction [11], cost, efficiency, reliability or some other
requirements.
The design process can be detailed and broken down in a number of consequential steps or/and
loops. By giving a shot overview, the size, shape and material of the core must be chosen, and the
number of turns of the primary and secondary windings. The size of the wire and the insulation
determines if the windings will fit in the space available. The windings must be arranged for
minimum leakage flux. The insulation determines the permissible temperature rise, and means of
cooling must balance the loss of energy under load. The number of primary turns is determined so
that the magnetizing current is limited to an acceptable value, and depends on the length of the
core. The area of the core is determined by the required volts per turn, now that the total number
of turns is known. These are the things that determine the size and weight of a transformer. Design
process is an iterative process that may include a number of sweeping statements or simplifying
assumptions e.g. the flux density is uniform throughout the core.
The next lines give an overview of functionality parameters of a transformer, investigate the size of
a transformer and describe the model of a transformer.
3.1 Functionality specification
Despite the many types of electronic transformers, their theory of operation does not differ.
Electrical functions are usually similar but design characteristics can differ in certain ways. Some
examples are; unipolar versus bipolar core utilization, saturating or not saturating, degree of energy
storage, regulation, and transformer impedance.
The purpose of a transformer is a power conditioning where the voltage change-ratio and the
galvanic separation between the source and the load are achieved. The transformer can stand alone
for power conditioning or include more components in order to adapt the power source to the
load. Apart from that the power conditioning circuit may include protection circuits or elements
built into the transformer.
3. Design of a transformer
8
Rated power
The apparent power of a transformer depends on the flow densities such as electric current,
magnetic flux and heat loss power density [11]. Basically the size of transformer can be evaluated
according the rated power, frequency and the dissipated losses.
Rated primary and secondary voltage
The primary voltage and the power conditioning specify the number of turns in the windings.
Regulation
Regulation includes both the input and the output voltage variation of the transformer. Line
Regulation is the variation of main voltage, applied to the primary winding of the transformer. The
changing of voltage on the primary winding results in a change of a voltage on secondary side with
the same percent (which is by standard ±10%). Load Regulation is a change in output voltage with
a change of load.
δU 2 =
U 20 − U 2 L
⋅100%
U 2L
( 3.1 )
where the output voltage at no load is U20 and output voltage at full load is U2L.
Mounting and cooling conditions
The size of any electromagnetic or mechanic converter is basically limited by thermal conditions.
An allowed dissipation power can be given or the cooling conditions can be specified. One of these
conditions can specify the temperature in the transformer in respect to the generated heat losses.
The hot spot temperature and the allowed temperature design is the actual limit.
Insulation class
Insulation systems include all isolation materials used in manufacture of transformers such as
bobbins, sheet film, film tape insulations, varnish covering of wires, varnish for impregnation etc.
The temperature stability each component should not be lower than the temperature value for
specific insulation classes. The other important function is the electrical insulation and galvanic
separation.
Q.6
Does output voltage U2 decreases proportionally with the increase of the load and why?
How it depends on the size of a transformer?
3.2 Geometric modelling
The proportions of the core-type and the shell-type transformers are investigated. The purposes of
these investigations are
1. to determine theoretically the maximum flow cross-section areas for the electric and the
magnetic circuit,
2. to express the size expression based to transparent power, volume and the electromagnetic
flow densities,
3. to find out the proportions of the transformer when considering the heat losses and heat
transfer i.e. the maximum temperature rise in a transformer.
3. Design of a transformer
9
Geometry
The geometric parameterization of the shell-type and the core-type of transformer are visualized in
Figure 3.1. As it is expressed the transformers should fit in to the same rectangular volume.
Figure 3.1 The geometry parameterization of the shell type (left column) and the core type (right
column) of transformer. The upper row shows the magnetic flux flow plane (xy-plane) and
the lower row shows the electric current flow plane (xz-plane).
The overall transformer length ltr, width wtr and height htr defines the bounding volume Vtr that is a
volume of a rectangular parallelepiped box where the transformer fits in. The overall dimensions
can be expressed through a single dimension and proportions to this chosen reference dimension
such as relative width kW and relative height kH compared to the overall length ltr.
Vtr = ltr wtr htr = ltr ⋅ kW ltr ⋅ k H ltr = kW k H ltr
3
( 3.2 )
The length of the leg in the magnetic core is expressed by using the relative slot length ks and the
length needed to form a one electromagnetic pole. For the single-phase core and shell type of
transformers the number of poles Np=2.
lc =
ltr
(1 − k s )
Np
( 3.3 )
Similarly the length of the slot is formulated according to the length of a single electromagnetic pole
and the relative slot length.
ls =
ltr
ks
Np
( 3.4 )
The width of the slot is given by the total width of the transformer wtr minus the width of the
magnetic core yokes. The width of the magnetic back-core equals to the length of the magnetic legcore. Depending on the type of transformer core the width of the magnetic back core (yoke) is
different and this is taken into consideration with the width factor of a transformer kTW.
3. Design of a transformer
10
ws = wtr − kTW lc = kW ltr − kTW lc
( 3.5 )
The height of the magnetic core is given by the total height of the transformer htr minus the height
of the ends of windings. The height of the ends of windings equals to the length of the slot.
Depending on the type of transformer core the height of the ends of windings is different and this
is taken into consideration with the height factor of a transformer kTH.
hc = htr − kTH l s = k H ltr − kTH l s
( 3.6 )
The winding window or the slot is the available area inside the ferrous core where the winding
wires into.
⎞
ltr ⎛⎜
l
k s kW ltr − kTW tr (1 − k s )⎟ =
⎟
N p ⎜⎝
Np
⎠
⎞
k f ks ⎛
⎜ kW − kTW (1 − k s ) ⎟
= k t ltr2
⎟
N p ⎜⎝
Np
⎠
Ae = k t k f l s ws = k t k f
( 3.7 )
The available cross-section area for the ferrous lamination stack is
⎞
⎛
ltr
(1 − k s )⎜⎜ k H ltr − kTH ltr k s ⎟⎟ =
Np
Np ⎠
⎝
k (1 − k s ) ⎛⎜
k k ⎞
k H − TH s ⎟
= ltr2 l
N p ⎜⎝
N p ⎟⎠
Am = k l lc hc = k l
( 3.8 )
The product of cross-section areas to the bounding volume of a transformer indicates the
geometrically optimal relation between the electric and the magnetic core.
ltr k p =
k t k f k l (1 − k s )k s
Ae Am
= ltr
2
Vtr
kW k H N p
⎛
⎞⎛
⎜ kW − kTW (1 − k s ) ⎟⎜ k H − kTH k s
⎜
⎟⎜
Np
Np
⎝
⎠⎝
2
0
2
2
( 3.9 )
core type
shell type
8
⎞
⎟
⎟
⎠
2
2
2
6
4
2
4
12
6
10
6
2
4
0
0
10
0.5
1
2
relative transformer width, kW [-]
4
8 0.25
0.5
4
14
6
2
0
0.5
0.25
0.25
2
8
14
8
4
4
2
8
1
2
0
6
2
10
relative transformer height, kH [-]
4
4
4
1
12
10
8
4
6
4
2
relative transformer width, kW [-]
Figure 3.2 The geometrical optimum for a single-phase shell and core type of transformer by considering
the constant envelope volume.
2
4
2
8
3. Design of a transformer
11
Figure 3.3 The geometry of a single-phase shell and core type of transformer at the point of the
geometrical optimum.
Mean length of magnetic path for the shell-type and the core-type of transformer are
⎞
⎛l
l
⎛l
⎞
2⎜ tr − 0.5lc ⎟ + 2(wtr − 0.5lc ) = 2⎜ tr − tr (1 − k s ) + k wltr ⎟ =
⎟
⎜ 2 Np
⎠
⎝2
⎠
⎝
⎛1
1 − k s ⎞⎟
= 2ltr ⎜ + k w −
⎜2
N p ⎟⎠
⎝
( 3.10 )
⎞
⎛
l
l
l
2(ltr − ls − lc ) + 2(wtr − lc ) = 2⎜ ltr − tr (1 − k s ) − tr k s + k wltr − tr (1 − k s )⎟ =
⎟
⎜
Np
Np
Np
⎠
⎝
( 3.11 )
⎞
⎛ 2 − ks
= 2ltr ⎜1 −
+ kw ⎟
⎟
⎜
N
p
⎠
⎝
Mean length of the winding for the shell-type and the core-type of transformer are
⎞
⎛
l
l
l
2(ltr − ls − lc ) + 2(htr − ls ) = 2⎜ ltr − tr (1 − k s ) − tr k s + k hltr − tr k s ⎟ =
⎜
Np
Np
N p ⎟⎠
⎝
⎞
⎛ 1 + ks
= 2ltr ⎜1 −
+ kh ⎟
⎟
⎜
Np
⎠
⎝
⎞
⎛l
l
⎞
⎛l
2⎜ tr − 0.5ls ⎟ + 2(htr − 0.5ls ) = 2⎜ tr − tr k s + k hltr ⎟ =
⎟
⎜ 2 Np
⎝2
⎠
⎠
⎝
⎛1
k ⎞
= 2ltr ⎜ + kh − s ⎟
⎜2
N p ⎟⎠
⎝
Q.7
( 3.12 )
( 3.13 )
What would be different if the geometric optimum would be derived for the three-phase
transformer with different core types?
3. Design of a transformer
12
3.3 Size equations
The size of transformer is related to the power of the electromagnetic device and the allowed flow
densities such as current density, flux density and the loss density.
Ideal transformer
In an ideal transformer the magnetic coupling between the windings is perfect. There is the same
core flux φ(t) that links each turn of each winding. Apparent power of an ideal lossless transformer
is expressed as
1
S = UmIm
2
( 3.14 )
here is assumed sinusoidal variation of voltage and current and instead of rms values the peak
values has been used instead. Considering the lossless electromagnetic circuit the voltage of the
electric circuit can be directly linked to the induced back electromotive force (emf) and the
magnetic flux in the magnetic circuit.
u (t ) = U m cos(ω t ) = e(t ) =
dψ (t )
dφ (t )
=N
dt
dt
( 3.15 )
Ideally it is assumed that all the magnetic flux links with the winding and flux can be expressed
directly from the voltage that is applied to the winding.
φ (t ) = Φ m sin (ω t ) =
Um
sin (ω t ) = Bm Am sin (ω t )
ωN
( 3.16 )
The maximum value of magnetic flux Φm is related to the cross-section area of the pure magnetic
conductor Am and the maximum flux density Bm that is defined by the material ability to conduct
magnetic flux and the magnetic saturation Bsat . Similar to the magnetic circuit, the flow in the
electric circuit is defined by current Im (maximum value) in the single turn which is related to the
total Ampere turns NIm i.e. magnetomotive force (mmf), the cross-section area of the pure electric
conductor Ae and the maximum current density Jm that is defined by device’s ability to conduct heat
flow and the thermal limit ϑcoil.
i (t ) = I m cos(ω t + ϕ ) =
NI m
J A
cos(ω t + ϕ ) = m e cos(ω t + ϕ )
N
N
( 3.17 )
By substituting the maximum values of voltage and current in the equation of the apparent power
the size of the transformer can be expressed through the nominal power and the magnetization
frequency the allowed current and flux density.
1
1
S = U m I m = ω Bm J m Ae Am
2
2
( 3.18 )
The specific power of transformer depends on the electric loading Jm, the magnetic loading Bm and
the magnetization frequency ω as well as the geometry relates dimension ltr and proportions kp.
S 1
= ω Bm J m ltr k p
Vtr 2
( 3.19 )
The size of the transformer can be smaller by increasing the loadings or magnetization frequency.
These parameters are related to the power loss generation and therefore it is important to study the
transformers ability to dissipate the heat.
3. Design of a transformer
13
Considering losses and heat dissipation
The properties of media, where the flows take place, have not been considered so far. Apart from
the electric current flow and the magnetic flux flow there is also a heat flow that is caused by power
loss sources. The power loss sources are related to the electrical and magnetic loadings and the
material properties. The conductor loss for the direct current Pcu.dc is expressed through the power
loss density, which depends on resistivity ρ and the current density square J2, and the volume of the
conductor Ve.
2
Pcu .dc
1
⎛ JA ⎞ l N
= I R = ⎜ e ⎟ ρ e = ρ J 2 Ae le = ρ J m2Ve
Ae
2
⎝ N ⎠
N
2
( 3.20 )
The remagnetization loss in the magnetic conductor for the symmetric sinusoidal excitation can be
found from the specific loss data kfe and core volume Vm
Pfe = k fe (Bm , ω ) Vm
( 3.21 )
Alternatively, the total core loss can be found from the loss separation formula, where the
hysteresis, eddy current and anomalous loss coefficients khy, h, kec, kan complete the expression.
Pfe = (k hy Bmh ω + k ec Bm2 ω 2 + k an Bm1.5ω 1.5 )Vm
( 3.22 )
The relation between the flow density and the loss density in an electric conductor can be expressed
as
qe =
1 2
Jmρ
2
( 3.23 )
and the approximate relation in the magnetic conductor, where km indicates the loss coefficient.
qm =
1 2 2
Bm ω k m
2
( 3.24 )
According to the heat transfer there is a balance between the heat sources and the heat dissipation.
The steady state heat distribution is defined by the Poisson partial differential equation,
λx
d2
d2
d2
ϑ
+
λ
ϑ
+
λ
ϑ+q =0
x
x
dx 2
dx 2
dx 2
( 3.25 )
where thermal conductivities λx, λy and λz define the heat flow from a region with higher
temperature ϑ to a region with lower temperature, respectively, in x, y and z direction according to
the spatial heat source distribution q [W/m3] and predefined dissipation. In case of one-dimensional
(1D) heat distribution the heat is generated within the material itself is
λx
d2
ϑ+q=0
dx 2
( 3.26 )
By applying the boundary conditions that there are no heat flow Q [W/m2] through the symmetric
surface(s) and the predefined temperature of the bounding surfaces is ϑs, then the temperature
distribution in a plate with a thickness of d=2xs and the reference temperature ϑ(xs)=ϑs
ϑ (x ) = ϑ (xs ) +
(
q
2
⋅ xs − x 2
2λ x
)
And in a cylinder with a diameter of d=2rs and the reference temperature ϑ(rs)=ϑs
( 3.27 )
3. Design of a transformer
14
ϑ (r ) = ϑ (rs ) +
(
q
2
⋅ rs − r 2
4λr
)
( 3.28 )
ϑ=ϑs
d/2
ϑ=ϑs
0.5dmax Q=0
q,λ
Q=0
0.5dmin
Figure 3.4 The symmetric part of a conductor cross-section with an internal heat generation q and an
isotropic thermal conductivity λ. The symmetry planes with predefined heat flux Q are the
same for a cylinder and a plate.
The highest temperature rise Δϑ in the centre of a plate or a cylinder, which thickness or diameter,
respectively is denoted with d and the geometry related coefficient kg is 1 for a plate and 2 for a
cylinder.
Δϑ =
q d2
2k g λ 4
( 3.29 )
The realistic cross-section area of an electric conductor or a magnetic conductor is between the
rectangle and the circle, which corresponds to the striped area in Figure 3.4. The geometric shape
and proportions can be taken account with the geometry related coefficient that is found
empirically from the ratio of the shortest dimension to the longest dimension of the cross-section
area.
⎛d
k g = 1 + ⎜⎜ min
⎝ d max
⎞
⎟⎟
⎠
2
( 3.30 )
By choosing that the electric circuit produces the same amount of losses than the magnetic circuit,
and the total temperature rise in the electromagnetic device is the sum of the temperature rise in the
electric circuit and in the magnetic circuit.
⎛
⎞
⎜
⎟
⎜
⎟
2
2
min (l s , ws )
min (lc , hc )
q⎜
⎟ = qk
Δϑ =
+
ϑ
8 ⎜ ⎛ ⎛ min (l , w ) ⎞ 2 ⎞
⎛ ⎛ min (l , h ) ⎞ 2 ⎞ ⎟
s
s
c
c
⎜ λ e ⎜1 + ⎜
⎟ ⎟ λm ⎜1 + ⎜⎜
⎟ ⎟⎟
⎜ ⎝ max(lc , hc ) ⎟⎠ ⎟ ⎟
⎜ ⎜ ⎜⎝ max(l s , ws ) ⎟⎠ ⎟
⎠
⎝
⎠⎠
⎝ ⎝
( 3.31 )
Even though the thermal coupling between the electric and the magnetic circuits is complex, this
simple approach helps to choose the geometrical proportions of a transformer in order to reduce
the maximum temperature rise. A narrow and high cross-section of a circuit has a smaller
temperature rise than a square shaped cross-section with the same area. Therefore the results
3. Design of a transformer
15
(Figure 3.5) indicate the sum of the thermal loadings in the electric circuit and in the magnetic
circuit that have their maximum thermal resistance at the different geometrical proportions of a
transformer. In the sake of comparability the thermal conductivity is taken the same λe=
λm=1W/mK. The maximum thermal resistance for the electric circuit is in region kH=0.5 kW=0.7 –
shell and kH=0.25 kW=1.1 – core, and for the magnetic circuit is in region kH=1.1 kW=0.25 – shell
and kH=0.7 kW=0.5 – core. The thermal resistance is calculated in respect with the transformer
envelope volume that is Vtr=1 m3.
Rϑ =
kϑ
Vtr
( 3.32 )
core type
024
2
4
2
0 82 46
6
4
8
4
12
16
12
1
14
8
10
6
8
6
16
10
02 4
4
02
10
10
0.5
2
4
6
2
14
relative transformer height, kH [-]
4
2
6
4
2
4
0
shell type
8
0.25
0.25
0.5
1
2
4
8 0.25
relative transformer width, kW [-]
0.5
1
2
4
relative transformer width, kW [-]
Figure 3.5 Thermal resistance Rϑ·1000 in respect with type of transformer and proportions of
transformer geometry.
Q.8
How could be understood Faraday’s law, Lenz law and the definition and sign of the
inductance element in circuit theory according to Kirchhoff’s law?
Q.9
How could be understood the fundamental transformer equation
U rms = 4.44 Nfφmax and the equation derived above?
3.4 Equivalent circuit of a transformer
Equivalent circuit facilitates to analyze the electromagnetic behaviour of the transformer.
Ideal transformer
Ideal describes the ideal electromagnetic coupling with no losses or voltage/mmf drop in the
electric/magnetic circuits. The electrical equations in the ideal transformer are
dψ 1
dφ
⎧
u
=
=
N
1
1
⎪⎪
dt
dt
⎨
⎪u = dψ 2 = N dφ
2
⎪⎩ 2
dt
dt
( 3.33 )
8
3. Design of a transformer
16
The magnetic equations
⎧φ = PN1i1
⎨
⎩φ = PN 2 i2
( 3.34 )
where for the ideally coupled windings the primary mmf balances the secondary mmf. The mutual
flux φ is the means of transfer of energy from primary to secondary, and links both windings. In an
ideal transformer, this flux requires negligibly small ampere-turns to produce it, so the net ampereturns, primary plus secondary, is about zero. When a current is drawn from the secondary in the
positive direction, ampere-turns decrease substantially. This must be matched by an equal increase
in primary ampere-turns, which is caused by an increase in the current entering the primary in the
positive direction. In this way, the back-emf of the primary (the voltage induced in it by the flux φ)
equals the voltage applied to the primary, as it must. This fundamental explanation of the operation
of a transformer must be clearly understood before proceeding further. The transformation
coefficient n is in accordance with instantaneous power balance
n=
u 2 N 2 i1
=
=
u1 N1 i2
( 3.35 )
If an impedance Z2 is connected to the secondary then U2/I2=Z2 and considering the relations
between the primary and the secondary voltages and currents then the impedance Z1 by the
primary terminals is
Z1 =
1
U1 1 U 2
= 2
= 2 Z2
Ii n I2 n
( 3.36 )
that is the transformer property of changing impedances.
Real transformer
In the real transformer, the winding resistances are not zero, the magnetic coupling between the
coils is not perfect and the reluctance of the magnetic core is not zero. Including the imperfections
to the formulation of the ideal transformer the electrical equation becomes
di2
di1
⎧
⎪⎪u1 = R1i1 + L1 dt − M dt
⎨
⎪u = − R i − L di2 + M di1
2 2
2
⎪⎩ 2
dt
dt
( 3.37 )
magnetic equations
Lσ 1i1 Mi1 L1i1
⎧
⎪φ1 = φσ 1 + φ21 = N + N = N
⎪
1
2
1
⎨
⎪φ = φ + φ = Lσ 2 i2 + Mi2 = L2 i2
12
σ2
⎪⎩ 2
N2
N1
N2
(3.38 )
This treats the windings as a pair of mutually coupled coils with both primary and secondary
windings passing currents. The negative sign in these equations arise from the reversed direction of
the secondary current i2. From the transformer equation, the primary MMF must equal the
secondary MMF, and since these are in opposite directions, in an ideal transformer they cancel so
that there is no overall resultant flux in the core. That this is so can be seen by realising that any
unopposed primary emf would create a large primary current and therefore a large flux in the core
due to the primary winding. However, this large flux would necessarily cause a large current to flow
3. Design of a transformer
17
in the secondary circuit and this current must create an opposing flux that effectively cancels the
initiating primary flux. In a non-ideal transformer, the resultant flux in the core is that needed to
magnetise the core. This is called the magnetising flux. Apart from the coupling flux the total flux
linkage of the coil includes also a leakage flux that does not couple with the secondary coil. By
describing the flux current elements as inductances the imperfection of the real magnetic coupling
can be taken into the consideration.
N1
⎧
⎪ L1 = Lσ 1 + N M
⎪
2
⎨
⎪L = L + N 2 M
σ2
⎪⎩ 2
N1
( 3.39 )
after the substitutions of the ‘description’ of magnetic circuit into the electrical equations the system
equation becomes
di1 1
d
⎧
=
+
− M (i1 − ni2 )
u
R
i
L
σ
1
1
1
1
⎪⎪
dt n dt
⎨
⎪u = − R i − L di2 + M d (i − ni )
2 2
σ2
1
2
⎪⎩ 2
dt
dt
( 3.40 )
and the resulting equivalent circuit is shown in Figure 3.6.
Figure 3.6 Circuit representation for the equation system of a transformer
Figure 3.7 Complete equivalent circuit for a transformer includes the effects of shunt and inter-winding
capacitance, stray inductance, magnetic loss, and winding resistance. Under low frequency or
large-signal conditions, the shunt primary inductance can become nonlinear if the transformer
is driven into saturation.
3. Design of a transformer
18
Figure 3.7 indicates the complete equivalent circuit that is equally applicable to pulse and wideband
transformers [2][10]. This results that the circuit can be used in the analysis that focuses rather on
time domain (pulse transformers) rather than frequency domain (band transformers). The particular
difference between Figure 3.6 and Figure 3.7 is that the first circuit neglects the capacitance effects
between the turns and between the windings, magnetic non-linearity is ignored and iron losses are
ignored. For a given transformer, the values ofof many equivalent circuit elements are same for
pulse and wideband application. The circuit elements determine the electromagnetic behavior of the
transformers. Description of the circuit parameters:
•
Nonzero resistance of the windings (R1 and R2) power dissipation through heating, and
impedance transformation,
•
Frequency dependence of the material permeability μr=f(freq),
•
Magnetic losses (Rc), due to material opposition (friction loss) to orient domain walls
according to the external magnetizing field or due to material opposition (conductive loss)
to conduct current, which is caused by the induced voltage in the alternating magnetic field.
•
Intra-winding capacitance (turn to turn within a winding—C1 C2), Inter-winding
capacitance (primary to secondary—C12) Parasitic capacitances limit the upper bandwidth
of operation and also reduce the isolation the transformer can provide
•
Finite primary winding inductance (Lm), For large signal operation, the core will saturate,
and the inductance will change during the course of a voltage cycle. This causes non-linear
behavior, and can lead to catastrophic transformer failure. It is important to realize that the
saturation problem is a function of the applied voltage and frequency only.
•
Finite flux capability of the core material, leading to saturation (non-linear behavior of Lm)
•
Leakage inductance of the windings (L1 and L2)
•
The equivalent circuit consists of an ideal transformer of ration 1:n that ideally establish the
mutual flux,
In the ideal transformer the magnetization is lossless
E1 = ωN1Φ
( 3.41 )
in the real transformer the magnetic core has a mmf-drop that drags a magnetizing current
I 0m =
ωN1
Xm
Φ
( 3.42 )
In addition the magnetization process causes losses due to magnetic hysteresis in the materials and
induced electromotive force that is reason for the eddy currents in a conductive material.
I 0c =
ωN1
Rc
Φ=
Pfe
ωN1Φ
( 3.43 )
The equivalent circuit can be further simplified by allowing small calculation inaccuracies without
significantly increasing the computation error.
ƒ
The small voltage drop in R1 and X1 compared to U1, and Small magnetizing current I0
in comparison with load current I1 allows the shunt terminals transfer to primary
terminals,
ƒ
The secondary quantities R2 and X2 may be replaced on the primary side by using ideal
transformer property,
3. Design of a transformer
ƒ
19
The new equivalent circuit can be used for small power transformer that simplifies the
analysis and parameter identification from the experiments.
Phasor diagram
With steady state a.c. conditions the instantaneous values can be represent with phasors in phasordiagrams (frequency domain). The best way to understand things is to draw a phasor diagram of the
currents, voltages and fluxes. In doing this, we draw the diagram in stages, showing how to proceed
step by step so that it would be possible to make accurate numerical estimates of each quantity. The
primary current is given by magnetizing current and the referred secondary current for a loaded
transformer:
I1 = I 0 + nI 2
( 3.44 )
The secondary voltage can be expressed from the equivalent circuit
U 2 = E2 − I 2 (R2 + jX 2 ) = nU 1 − nI1 (R1 + jX 1 ) − I 2 (R2 + jX 2 )
U1
Im
Im
E2
jx1I1
R1I1
E1
( 3.45 )
I1
R2I2
I2’
U2
I0
I2
Re
Re
Φ
Φ
Figure 3.8 Phasor diagrams of the operating frequency domain for the primary side (on the left) and for
the secondary side (on the right).
The design goal is to design a transformer as close to an ideal transformer as possible. Therefore
some of the transformers such as measuring instrument transformers need to be carefully designed.
4 Numerical field modelling of a transformer
Most of the Finite Element computational software for electromagnetism considers current-driven
magnetic problems. However, the resistance and the linking flux of the current carrying coil can be
evaluated, which in turn is the basis to estimate the voltage across the winding. In the case of
determining of an operating point of a transformer a voltage across the primary and the secondary
load impedance of the transformer is the input to iteratively determine the currents required to
produce specified terminal voltages with arbitrarily specified external loads. Apart from the
magnetic problem the thermal problem is present. According to the cooling conditions and the hotspot temperature of the maximum allowed currents can be evaluated. When using FEMM for 2D
FE solver to compute 3D problem in electromagnetism the following effects can be considered:
•
The effect of the small air gaps at lamination joints,
•
Rolling direction by introducing anisotropic material,
•
End turn impedance,
•
Hysteresis and eddy current loss in the laminations,
•
Proximity and skin effects in the transformer windings
A simple transformer model has been used to visualize the example of determining the operation
point of transformer at an open-circuit, a nominal loading and also a short-circuit condition. The
static model in electromagnetism is established i.e. all the reference and calculated values can be
seen as the maximum values. This mistake is allowed in order to facilitate the understanding of FE
modelling.
4.1 Model parameterization
The example transformer consists of a primary winding (orange rectangles in the left) and a
secondary winding (yellow rectangles) that occupy the slot of the UI or LL transformer core (a grey
square with an opening). The height of the square shaped iron core is 40 mm and this dimension is
kept constant while the width of the core sides is to be changed in order to find the optimal
proportions between the iron core and the windings (Figure 4.1).
ϑ=75°C
coilP
coilS
coilP
coilS
A=0Vs/m
Figure 4.1 FE model setup for solving the problem of heat transfer (with Mirage) and electromagnetism
(with FEMM)
4. Numerical field modelling of a transformer
22
The initial dimensions are the width of the winding ww=10 mm and the height of the winding
wh=20 mm, which define the cross-section area of the primary and the secondary windings. The
width of the iron core is cw=10 mm and the region around the transformer is defined by two
dimensions aw=20 mm and ah=20 mm. These last dimensions are necessary for Finite Element
Modelling of the electromagnetic problem. The length of the transformer along the z-axis is cl=40
mm.
Apart from the geometry the necessary information is the copper fill-factor of the winding Kf=0.6,
that specifies the equivalent thermal conductivity λeqv=0.2 W/Km and the iron lamination that has
iron fill-factor of lamf=0.95 and the thickness of the lamination lamd=0.5 mm that has the thermal
conductivity of λfe=24 W/Km. The maximum allowed temperature in the winding is ϑcoil=135°C
and the temperature rise in the device is Δϑ =60°C. The number of turns in the primary winding is
N1=1 turn and the turn ratio is N1/N2=1. Therefore total ampere turns and voltage per turn are
considered. The initial current density is 2 A/mm2 and this does not correspond always to the
thermal condition therefore the current density has to be found iteratively from the thermal model
that consider the geometrical change and the copper and the core losses in the loaded transformer.
The specific copper losses are found in respect with current density square, temperature dependent
resistivity and the copper fill factor. The iron losses that can be taken into consideration with the
lag angle between the magnetic field intensity H and the flux density B and core conductivity are
introduced. In this example the frequency is set freq=0 Hz and the core losses are estimated from
the polynomial coefficient that fit the material characteristic (Figure 4.2)
2
1.8
flux density B, [T]
1.6
1.4
1.2
1
0.8
0.6
0.4
M 330 - 50 A
M 530 - 50 A
0.2
0
2
10
3
10
4
10
magnetic field intensity H, [A/m]
M 330 - 50 A
M 530 - 50 A
-2
10
-1
10
0
10
specific losses pc, [W/kg]
M 330 - 50 A
M 530 - 50 A
1
10
0
10
1
10
2
10
specific apparent power pc, [VA/kg]
Figure 4.2 Characteristics of the electromagnetic steel type, which are used in the simulation and
prototyping.
In the example model the chosen type of electromagnetic steel is M 530 – 50 A.
In the next lines the example of the main model parameterization is given. The text can be copied
into lua file, and can be executed together with additional sub-files in order to solve a problem on
heat transfer or electromagnetism. The variable that makes the choice between heat transfer
problem setup (that is solved in Mirage) and the task on electromagnetism, which is solved in
FEMM is ask_fem. Depending on the switch 0 or 1 the main lua script suppose to be executed in
FEMM or Mirage, respectively.
A main command routine can be made that makes selection between the software and the problem
to be solved.
4. Numerical field modelling of a transformer
23
-- magnetic and thermal design of a transformer
-- selection
ask_fem=1;
-- dofile("tmp_select_1.lua");
-- 1-heat transfer, 0-electromagnetism
-- read in from a file that is created before
-- device parameterization
U1p=230*sqrt(2);
-- primary voltage max value
Bcm=1.5;
-- initial flux density max value
Jcm=2;
-- initial current density max value
Tcoil=135;
-- max coil temperature
DeltaT=60;
-- temperature rise in the device
Kf=0.6;
-- copper fill factor in the winding
rcu0=2.4e-8;
-- copper resistivity at 20C
rcu=rcu0*(1+0.004*(Tcoil-20)); -- copper resistivity at Tcoil
N1=1;
-- number of turns in the primary side
Nratio=1;
-- turn ratio in the transformer
lam_d=0.5;
-- thickness of lamination
lam_f=0.95;
-- lamination fill factor
lam_loss=15;
-- loss angle
labmda_eqv=0.2;
-- equivalent thermal conductivity of a coil
labmda_fe=24;
-- thermal conductivity of a iron core
labmda_air=0.03;
-- thermal conductivity of air
freq=0;
-- frequency
pfe=(2.5517*Bcm^2-1.2936*Bcm+0.8143)*7700 -- core loss density W/m3 at 50Hz
-- geometry parameterization
cl=40e-3;
ww=10e-3;
wh=20e-3;
cw=10e-3;
aw=20e-3;
ah=20e-3;
-------
length of core in z-axis
width of the winding
height of the winding
width of the iron core
width of air region from the winding
height of air region from the core
if ask_fem==0 then
h=openfile("tmp_magn_res.txt","w")
closefile(h);
end
if ask_fem==1 then
h=openfile("tmp_heat_res.txt","w")
closefile(h);
end
-- for kh = 1,9,1 do
for kh = 5,5,1 do
-- geometric change
ww=kh*2e-3;
wh=kh*4e-3;
cw=20e-3-ww;
-- clear content of the output file
-- clear content of the output file
-- multiple step
-- single step
-- width of the winding
-- height of the winding
-- width of the iron core
-- magnetization (idling transformer)
if ask_fem==0 then
filename=format('%s%g%s','tmp_trafo_B_',kh,'.bmp') -- format filename
dofile("PTC_magn_1.lua");
-- execute a lua script
mo_close(); mi_close();
-- close pre and post-processor
end
-- temperature limit of the loaded transformer
if ask_fem==1 then
filename=format('%s%g%s','tmp_trafo_T_',kh,'.bmp') -- format filename
dofile("PTC_heat_1.lua");
-- execute a lua script
ho_close(); hi_close();
-- close pre and post-processor
end
end
4. Numerical field modelling of a transformer
24
Q.10 The core loss model in the example bases on polynomial fit of the loss characteristic at a
single frequency, how do consider harmonic losses? Are the core losses over or
underestimated if they are calculated according to superposition of flux density
frequency spectrum for the saturated core?
Q.11 What is the purpose of specific apparent power characteristics and how are these
connected to magnetization characteristics?
4.2 Magnetization
This sub-section describes an iterative model to estimate magnetization current of a transformer/an
inductor when the target flux density is given. First a theoretical introduction will be given, that is
later followed with an undemanding example, which is the sub-file PTC_magn_1.lua that is called
from the model parameterization file.
Transformer can be seen as an electromagnetic inductor that is loaded with a secondary winding.
Therefore, when the secondary loading is absent the magnetization process of the transformer can
be studied. When considering an ideal electromagnetic inductor then the losses in the electric circuit
and in the magnetic circuit can be neglected. According to Ampere’s and Faraday’s law the current
carrying conductor causes magnetic field and alternating magnetic field induces an opposing
electromagnetic force, respectively. For pure inductance the applied voltage equals to the
transformed induced electromotive force and the latter is π/2 ahead of the magnetic flux, the flux
linkage and the current. The magnetic flux depends on magnetic permeance of an average flux path
and the magnetomotive force.
Hl =
B
μ
l =φ
l
1
= φ R = φ = NI
P
μA
( 4.1 )
The corresponding magnetic equivalent circuit including core, leakage and gap permeances is
expressed in Figure 4.3. The sum of the magnetizing flux φc in the core and the leakage flux φσ
define the total fluxφ.
Pc2
φσ
Pσ
Pc1
Pgap
NI
φc
Figure 4.3 The magnetic equivalent circuit of an electromagnetic inductor. The magnetic core is divided
into two parts: one which is enclosed with winding and the rest. The gap permeance indicates
the total distance between the laminates on the same layer.
The nonlinear permeance, Pc1, indicates the electromagnetic core that is enclosed with the winding.
The rest of the magnetic circuit is noted with Pc2. The magnetic/mechanical contact between the
laminations at the same plane may include a small air-gap that can be taken into consideration with
Pgap. The linear leakage, Pσ, can be geometrically derived as it is shown for the symmetric part of
the winding/core cross-section in Figure 4.4.
4. Numerical field modelling of a transformer
25
φc
φc
φσn
φn
0.5Rcn
φn
0.5Rσh
φσ2
φ2
φσ1
φ1
0.5nni
φ2
0.5n2i
0.5Rc2
0.5Rσ2
0.5Rσ1
0.5n1i
0.5Rc1
Figure 4.4 A symmetric part of a coil on an iron core and the corresponding equivalent circuit.
When the magnetic material BH curve has hysteresis, the fundamental components of vectors H
and B are out of phase. In any case the induced voltage is π/2 ahead of the flux, but due to losses a
current component appears that is in phase with induced voltage and the corresponding active
power covers the hysteresis loss in the magnetic material. Therefore hysteresis loss can be
introduced by defining an angle between the H and B vectors, which is he case in FEMM AC
problem of electromagnetism. In case of alternating magnetic field and the conductive magnetic
core the induced electromagnetic force causes currents. These eddy currents cause opposing field
and losses. The following assumption can be made in order to simplify the further calculations:
•
Full penetration of magnetic flux,
•
The magnitude of magnetic flux density is independent of the core loss mechanisms and
depends only on magnetizing current,
•
Uniform distribution of magnetic flux in order to facilitate core loss estimation.
As the geometry of the transformer is symmetrical, all the parameters, which are introduced for the
electrical equivalent circuit (Figure 3.6), can be obtained. Instead of voltages, currents, inductances,
etc the voltage per turn, the total current, the magnetic permeances (L/N2) can be introduced
instead. This facilitates to keep thinking rather in cross-section areas than a number of turns. The
corresponding electrical equivalent circuit is shown in Figure 4.5.
Lσ μAσ
=
N2
Lσ
U
N
RΩ 1 Lw
=
N 2 γ k f Aw
Pfe
Rc
=
2
N
(U N )2
Lm μAc
=
N2
Lc
IN = Jk f Aw
Figure 4.5 The electrical equivalent circuit of an inductor where the voltage and currents are represented
in voltage per turns and ampere turns.
4. Numerical field modelling of a transformer
26
The electrical equation of the circuit (Figure 4.5) is
Pfe
⎛
μA
⎜
jω σ
Lσ (U N )2
μA
U
⎜ Lw
= JAW k f ⎜
+ jω σ +
Pfe
N
Lσ
γk A
μAc
⎜ f w
+
j
ω
⎜
Lc
(U N )2
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎠
( 4.2 )
Instead of constant voltage the magnetizing current density is found iteratively with respect to
constant flux density. The lua script PTC_magn_1.lua consists of a transformer geometry setup, a
material characterization, a FE model setup, which includes boundary conditions and source
definitions, solver and result analysis. The post-processing routine calculates iteratively the
magnetization current density that grounds the desired flux density in the core. The stability and
converging speed is determined by weight variable.
-- solving a electromagnetism problem with FEMM 4.0
x0=0; y0=0;
-- origin and the rest of the coordinates
x1=x0+aw; x2=x1+ww; x3=x2+cw; x4=x3+ww;
x5=x4+ww; x6=x5+cw; x7=x6+ww; x8=x7+aw;
y1=y0+ah; y2=y1+cw; y3=y2+wh; y4=y3+cw; y5=y4+ah;
xi=1e-4; yi=1e-4;
-- displacement increment
-- additional parameters
ElemSize=0.001;
-- parameter calculation
Aw = ww*wh;
N2 = ceil(N1*Nratio);
towards infinity
I1m = (Jcm*1e+6)*Aw*Kf;
I2m = 0;
d1 = sqrt(4/pi*Aw*Kf/N1);
d2 = sqrt(4/pi*Aw*Kf/N2);
-- max finite element size
-- winding cross section
-- rounds the value to the nearest integer
-----
max value of the primary current
max value of the secondary current
diameter of primary conductor
diameter of secondary conductor
-- create a new preprocessor for a magnetics problem
create(0)
-- problem definition
mi_probdef(freq,"meters","planar",1e-8,cl)
-- boundary definitions
mi_addboundprop("tang",0,0,0,0,0,0,0,0,0) -- 'zero' magnetic potential
-- material definitions
mi_addmaterial("core",500,500,0,0,0,2,lam_d,lam_loss,lam_f,0)
mi_addmaterial("ins",1,1,0,0,0,0.002,0,0,1,0)
mi_addmaterial("air",1,1,0,0,0,0,0,0,1,0)
--mi_addmaterial("coilP",1,1,0,0,0,1/rcu,0,0,1,4,0,0,1,d1*1000)
--mi_addmaterial("coilS",1,1,0,0,0,1/rcu,0,0,1,4,0,0,1,d2*1000)
mi_addmaterial("coilP",1,1,0,0,0,1/rcu,0,0,1,0)
mi_addmaterial("coilS",1,1,0,0,0,1/rcu,0,0,1,0)
-- circuit definition
mi_addcircprop("coilP",I1m,0,1)
mi_addcircprop("coilS",I2m,0,1)
-- definition of non-linear magnetic characteristics
mi_addbhpoint("core",0,0)
mi_addbhpoint("core",0.03,20.0)
mi_addbhpoint("core",0.09,40.0)
mi_addbhpoint("core",0.21,60.0)
mi_addbhpoint("core",0.42,80.0)
4. Numerical field modelling of a transformer
27
mi_addbhpoint("core",0.69,100.0)
mi_addbhpoint("core",0.92,120.0)
mi_addbhpoint("core",1.06,140.0)
mi_addbhpoint("core",1.18,160.0)
mi_addbhpoint("core",1.26,180.0)
mi_addbhpoint("core",1.32,200.0)
mi_addbhpoint("core",1.35,220.0)
mi_addbhpoint("core",1.37,240.0)
mi_addbhpoint("core",1.39,260.0)
mi_addbhpoint("core",1.41,280.0)
mi_addbhpoint("core",1.43,300.0)
mi_addbhpoint("core",1.49,400.0)
mi_addbhpoint("core",1.55,600.0)
mi_addbhpoint("core",1.58,800.0)
mi_addbhpoint("core",1.60,1000.0)
mi_addbhpoint("core",1.62,1200.0)
mi_addbhpoint("core",1.64,1400.0)
mi_addbhpoint("core",1.66,1600.0)
mi_addbhpoint("core",1.67,1800.0)
mi_addbhpoint("core",1.68,2000.0)
mi_addbhpoint("core",1.70,2200.0)
mi_addbhpoint("core",1.72,2400.0)
mi_addbhpoint("core",1.73,2600.0)
mi_addbhpoint("core",1.74,2800.0)
mi_addbhpoint("core",1.75,3000.0)
mi_addbhpoint("core",1.8091,10000)
-- primary winding (into the plane) coilP
mi_addnode(x1,y2)
-- add a new node at x,y
mi_addnode(x2,y2)
mi_addnode(x2,y3)
mi_addnode(x1,y3)
mi_addsegment(x1,y2,x2,y2)
-- add a new line segment from x1,y1 to x2,y2
mi_addsegment(x2,y2,x2,y3)
mi_addsegment(x2,y3,x1,y3)
mi_addsegment(x1,y3,x1,y2)
mi_addblocklabel(x1+xi,y2+yi)
-- add a new blocklabel at x,y
mi_selectlabel(x1+xi,y2+yi)
-- select the blocklabel closest to x,y
mi_setblockprop("coilP",0,ElemSize,"coilP",0,1,N1) -- region properties
mi_clearselected()
-- clear previous selections
-- primary winding (from the plane) coilP
mi_addnode(x3,y2)
-- add a new node at x,y
mi_addnode(x4,y2)
mi_addnode(x4,y3)
mi_addnode(x3,y3)
mi_addsegment(x3,y2,x4,y2)
-- add a new line segment from x1,y1 to x2,y2
mi_addsegment(x4,y2,x4,y3)
mi_addsegment(x4,y3,x3,y3)
mi_addsegment(x3,y3,x3,y2)
mi_addblocklabel(x3+xi,y2+yi)
-- add a new blocklabel at x,y
mi_selectlabel(x3+xi,y2+yi)
-- select the blocklabel closest to x,y
mi_setblockprop("coilP",0,ElemSize,"coilP",0,1,-N1) -- region properties
mi_clearselected()
-- clear previous selections
-- iron core
mi_addnode(x2,y1)
-- add a new node at x,y
mi_addnode(x6,y1)
mi_addnode(x6,y4)
mi_addnode(x2,y4)
mi_addsegment(x2,y1,x6,y1)
-- add a new line segment from x1,y1 to x2,y2
mi_addsegment(x6,y1,x6,y4)
mi_addsegment(x6,y4,x2,y4)
mi_addsegment(x2,y4,x2,y1)
mi_addblocklabel(x2+xi,y1+yi)
-- add a new blocklabel at x,y
mi_selectlabel(x2+xi,y1+yi)
-- select the blocklabel closest to x,y
mi_setblockprop("core",0,ElemSize,"none",0,1,0) -- region properties
mi_clearselected()
-- clear previous selections
4. Numerical field modelling of a transformer
28
-- primary winding (into the plane) coilS
mi_addnode(x4,y2)
-- add a new node at x,y
mi_addnode(x5,y2)
mi_addnode(x5,y3)
mi_addnode(x4,y3)
mi_addsegment(x4,y2,x5,y2)
-- add a new line segment from x1,y1 to x2,y2
mi_addsegment(x5,y2,x5,y3)
mi_addsegment(x5,y3,x4,y3)
mi_addsegment(x4,y3,x4,y2)
mi_addblocklabel(x4+xi,y2+yi)
-- add a new blocklabel at x,y
mi_selectlabel(x4+xi,y2+yi)
-- select the blocklabel closest to x,y
mi_setblockprop("coilS",0,ElemSize,"coilS",0,1,N2) -- region properties
mi_clearselected()
-- clear previous selections
-- primary winding (from the plane) coilS
mi_addnode(x6,y2)
-- add a new node at x,y
mi_addnode(x7,y2)
mi_addnode(x7,y3)
mi_addnode(x6,y3)
mi_addsegment(x6,y2,x7,y2)
-- add a new line segment from x1,y1 to x2,y2
mi_addsegment(x7,y2,x7,y3)
mi_addsegment(x7,y3,x6,y3)
mi_addsegment(x6,y3,x6,y2)
mi_addblocklabel(x6+xi,y2+yi)
-- add a new blocklabel at x,y
mi_selectlabel(x6+xi,y2+yi)
-- select the blocklabel closest to x,y
mi_setblockprop("coilS",0,ElemSize,"coilS",0,1,-N2) -- region properties
mi_clearselected()
-- clear previous selections
-- background air-region
mi_addnode(x0,y0)
-- add a new node at x,y
mi_addnode(x8,y0)
mi_addnode(x8,y5)
mi_addnode(x0,y5)
mi_addsegment(x0,y0,x8,y0)
-- add a new line segment from x1,y1 to x2,y2
mi_addsegment(x8,y0,x8,y5)
mi_addsegment(x8,y5,x0,y5)
mi_addsegment(x0,y5,x0,y0)
mi_addblocklabel(x0+xi,y0+yi)
-- add a new blocklabel at x,y
mi_selectlabel(x0+xi,y0+yi)
-- select the blocklabel closest to x,y
mi_setblockprop("air",0,ElemSize,"none",0,1,0) -- region properties
mi_clearselected()
-- clear previous selections
-- specify boundary conditions for the bounding air-region
mi_selectsegment(0.5*(x0+x8),y0) -- select the line closest to x,y
mi_selectsegment(x8,0.5*(y0+y5))
mi_selectsegment(0.5*(x0+x8),y5)
mi_selectsegment(x0,0.5*(y0+y5))
mi_setsegmentprop("tang",0,0)
-- set the line segment properties
mi_clearselected()
-- clear previous selections
mi_zoomnatural()
-- view all geometry
mi_saveas("tmp_magn_ini.fem")
mi_createmesh()
mi_analyse()
mi_loadsolution()
-----
save the file
create a mesh
solve the problem
load the solution
-- post-processing
-- iterative computation of magnetizing current density
weight=-5;
-- weight of iteration
nriter=10;
-- max number of iterations
err=1; hh=0;
-- initial conditions
while err==1 do
-- iterative loop
hh=hh+1;
if hh==1 then jc2=Jcm; else jc2=jc1; end
-- calculate the linking flux with core
4. Numerical field modelling of a transformer
mo_seteditmode("contour")
-- set to a contour mode
mo_addcontour(x5,0.5*(y2+y3))
-- left side of the core
mo_addcontour(x6,0.5*(y2+y3))
-- right side of the core
linere, lineim, advre, advim = mo_lineintegral(0) -- line integral
mo_clearcontour()
-- clear a previously defined contour
Bca=abs(advre)
-- give a new value to current density
if Bca-Bcm < -0.12 then jc1=jc2*2;
elseif Bca-Bcm > 0.12 then jc1=jc2*0.5;
else
jc1=jc2+weight*(Bca-Bcm);
end
I1m = (jc1*1e+6)*Aw*Kf;
-- max value of the primary current
mi_deletecircuit("coilP")
mi_addcircprop("coilP",I1m,0,1)
mo_close()
mi_saveas("tmp_magn_ini.fem")
-- save the file
mi_createmesh()
-- create a mesh
mi_analyse()
-- solve the problem
mi_loadsolution()
-- load the solution
mo_showdensityplot(0,1,Bcm,0.0,"real") -- show the flux density plot
Terr=abs(Bca-Bcm);
-- conditions to finish the iterative loop
if Terr<0.005 then err=0; end
if hh>nriter then err=0; end
-- messagebox(Bca)
-- messagebox(jc1)
end
-- block values
mo_showdensityplot(1,1,Bcm,0.0,"real") -- show the flux density plot
mo_savebitmap(filename)
-- saves the current view to specified file
mo_seteditmode("area")
mo_selectblock(0.5*(x1+x2),0.5*(y2+y3))
Iw1=mo_blockintegral(7)
-- total current
Aw1=mo_blockintegral(5)
-- block cross-section area
mo_clearblock()
mo_selectblock(0.5*(x2+x3),0.5*(y2+y3))
Pfe=mo_blockintegral(3)
-- hysteresis and lamination losses
Afe=mo_blockintegral(5)
-- block cross-section area
mo_clearblock()
-- circuit parameters
c_re, c_im, v_re, v_im, f_re, f_im=mo_getcircuitproperties("coilP")
Ipa=sqrt(c_re^2+c_im^2); Vpa=sqrt(v_re^2+v_im^2); Psip=sqrt(f_re^2+f_im^2);
c_re, c_im, v_re, v_im, f_re, f_im=mo_getcircuitproperties("coilS")
Ina=sqrt(c_re^2+c_im^2); Vna=sqrt(v_re^2+v_im^2); Psin=sqrt(f_re^2+f_im^2);
-- calculate the linking flux with core
mo_seteditmode("contour")
-- set to a contour mode
mo_addcontour(x5,0.5*(y2+y3))
-- left side of the core
mo_addcontour(x6,0.5*(y2+y3))
-- right side of the core
linere, lineim, advre, advim = mo_lineintegral(0) -- line integral
mo_clearcontour()
-- clear a previously defined contour
-- accumulate the main results
h=openfile("tmp_magn_res.txt","a") -- open an output file for writing
write(h,Ipa," ",Ina," ",Vpa," ",Vna," ",Psip," ",Psin," ",linere," ",advre,"
",jc1," ",jc2,"\n") – OBS! at same line
closefile(h)
-- close the output file for writing
29
4. Numerical field modelling of a transformer
30
4.3 Loaded transformer
By loading a transformer, the load current causes an opposing field to magnetizing field. Due to
that the magnetic flux and also the induced voltage decreases. At the same time the seeming
reduction is covered by an increased primary current, which in turn re-establishes the magnetizing
field and sets the induced voltage up to the same level. The transformer operation point can be
estimated according to the electrical equivalent circuit. The elements can be obtained from the FE
model of a transformer at each operating frequency:
•
Primary inductance, L1 =
∫ A JdS
z
( 4.3 )
I1 =1 A, I 2 =0 A
Sp
•
Secondary inductance, L2 =
∫ A JdS
z
( 4.4 )
I1 = 0 A, I 2 =1 A
Ss
•
Mutual inductance, L1 + L2 + 2 M =
∫ A JdS
z
I1 =1 A, I 2 =1
( 4.5 )
S p +S p
In the previous expressions, Az stands for the magnetic vector potential and J for the current
density. The surface integral is taken over an excited primary winding, an excited secondary winding
or, respectively, both of them.
The important limit of the electromagnetic loading is the temperature limit and hot-spot
temperature estimation, which will be discussed in the next section.
Q.12 How is related magnetic saturation, leakage flux and core losses to the electrical loading
of the transformer?
Q.13 How is related magnetic saturation, leakage flux and core losses to the magnitude and
frequency change of the primary voltage?
4.4 Temperature rise
The model setup of the problem on heat transfer is relatively inexperienced (Figure 4.1). First of all
the 2D model can consider heat dissipation all the surfaces that are perpendicular to the cross-section
of the electromagnetic device. Nevertheless, only three of these surfaces are considered as a subject
of heat dissipation. Instead of using heat transfer coefficient, which can take describe heat
dissipation by convection, the surface temperature is defined. This kind of boundary condition
disregards the actual cooling condition it rather states that one way or another i.e. normal or forced
cooling, the surface temperature has to be kept to the defined value.
Due to geometrical simplification the main insulation or the bobbin is ignored. Even though the
simplification is acceptable in the computation of the problem on electromagnetism, the nonattendance has significant importance in the thermal computation. In the present case it can be
assumed that the main insulation and the winding have the same equivalent heat transfer coefficient
and the larger loss area can consider the heat transport from the end turns. It is considered that
winding has much better heat transfer coefficient in the direction of current flow and worse in any
other directions.
From thermal design point of view:
•
The space for electric circuits is well utilized if they operate at the same loss density,
•
Transformer’s efficiency is largest when the copper losses equal to core losses,
4. Numerical field modelling of a transformer
•
31
The thermal conductivity of a winding is a few orders lower than for laminated core, thus
the winding cross-section determines the loading of the transformer.
The lua script for the heat transfer problem PTC_heat_1.lua is quite similar to the command lines,
which handle the problem set up for electromagnetism. The new script consists also of a
transformer geometry setup, a material characterization, a FE model setup, which includes
boundary conditions and source definitions, solver and result analysis. The post-processing routine
calculates iteratively the maximum current density that corresponds to the maximum temperature in
the coil. The hot-spot temperature is considered to be on the line that divides the coil cross-section
into two halves. The stability and converging speed is determined by weight variable.
-- solving a heat problem with Mirage 1.0
x0=0; y0=0;
-- origin and the rest of the coordinates
x1=x0+aw; x2=x1+ww; x3=x2+cw; x4=x3+ww;
x5=x4+ww; x6=x5+cw; x7=x6+ww; x8=x7+aw;
y1=y0+ah; y2=y1+cw; y3=y2+wh; y4=y3+cw; y5=y4+ah;
xi=1e-4; yi=1e-4;
-- displacement increment
-- additional parameters
ElemSize=0.001;
-- parameter calculation
Aw = ww*wh;
pcu1=(Jcm*1e+6)^2*rcu*Kf;
pcu2=pcu1;
winding
-- max finite element size
-- winding cross section
-- copper loss density in the primary winding
-- copper loss density in the secondary
-- create a new pre-processor for a thermal problem
newdocument()
-- problem definition
hi_probdef("meters","planar",1e-8,cl)
-- boundary definitions
hi_addboundprop("conv",2,0,0,20,10,0)
hi_addboundprop("Tamb",0,Tcoil-DeltaT,0,0,0,0)
hi_addboundprop("insulation",1,0,0,0,0,0)
-- convection
-- cooling surface temperature
-- zero thermal flux
-- material definitions
hi_addmaterial("core",labmda_fe,labmda_fe,pfe)
hi_addmaterial("ins",0.12,0.12,0)
hi_addmaterial("air",labmda_air,labmda_air,0)
hi_addmaterial("coilP",labmda_eqv,labmda_eqv,pcu1)
hi_addmaterial("coilS",labmda_eqv,labmda_eqv,pcu2)
-- primary winding (into the plane) coilP
hi_addnode(x1,y2)
-- add a new node at x,y
hi_addnode(x2,y2)
hi_addnode(x2,y3)
hi_addnode(x1,y3)
hi_addsegment(x1,y2,x2,y2)
-- add a new line segment from x1,y1 to x2,y2
hi_addsegment(x2,y2,x2,y3)
hi_addsegment(x2,y3,x1,y3)
hi_addsegment(x1,y3,x1,y2)
hi_addblocklabel(x1+xi,y2+yi)
-- add a new blocklabel at x,y
hi_selectlabel(x1+xi,y2+yi)
-- select the blocklabel closest to x,y
hi_setblockprop("coilP",0,ElemSize,1) -- define the region properties
hi_clearselected()
-- clear previous selections
-- primary winding (from the plane) coilP
hi_addnode(x3,y2)
-- add a new node at x,y
hi_addnode(x4,y2)
32
4. Numerical field modelling of a transformer
hi_addnode(x4,y3)
hi_addnode(x3,y3)
hi_addsegment(x3,y2,x4,y2)
-- add a new line segment from x1,y1 to x2,y2
hi_addsegment(x4,y2,x4,y3)
hi_addsegment(x4,y3,x3,y3)
hi_addsegment(x3,y3,x3,y2)
hi_addblocklabel(x3+xi,y2+yi)
-- add a new blocklabel at x,y
hi_selectlabel(x3+xi,y2+yi)
-- select the blocklabel closest to x,y
hi_setblockprop("coilP",0,ElemSize,1) -- define the region properties
hi_clearselected()
-- clear previous selections
-- iron core
hi_addnode(x2,y1)
-- add a new node at x,y
hi_addnode(x6,y1)
hi_addnode(x6,y4)
hi_addnode(x2,y4)
hi_addsegment(x2,y1,x6,y1)
-- add a new line segment from x1,y1 to x2,y2
hi_addsegment(x6,y1,x6,y4)
hi_addsegment(x6,y4,x2,y4)
hi_addsegment(x2,y4,x2,y1)
hi_addblocklabel(x2+xi,y1+yi)
-- add a new blocklabel at x,y
hi_selectlabel(x2+xi,y1+yi)
-- select the blocklabel closest to x,y
hi_setblockprop("core",0,ElemSize,1) -- define the region properties
hi_clearselected()
-- clear previous selections
-- primary winding (into the plane) coilS
hi_addnode(x4,y2)
-- add a new node at x,y
hi_addnode(x5,y2)
hi_addnode(x5,y3)
hi_addnode(x4,y3)
hi_addsegment(x4,y2,x5,y2)
-- add a new line segment from x1,y1 to x2,y2
hi_addsegment(x5,y2,x5,y3)
hi_addsegment(x5,y3,x4,y3)
hi_addsegment(x4,y3,x4,y2)
hi_addblocklabel(x4+xi,y2+yi)
-- add a new blocklabel at x,y
hi_selectlabel(x4+xi,y2+yi)
-- select the blocklabel closest to x,y
hi_setblockprop("coilS",0,ElemSize,1) -- define the region properties
hi_clearselected()
-- clear previous selections
-- primary winding (from the plane) coilS
hi_addnode(x6,y2)
-- add a new node at x,y
hi_addnode(x7,y2)
hi_addnode(x7,y3)
hi_addnode(x6,y3)
hi_addsegment(x6,y2,x7,y2)
-- add a new line segment from x1,y1 to x2,y2
hi_addsegment(x7,y2,x7,y3)
hi_addsegment(x7,y3,x6,y3)
hi_addsegment(x6,y3,x6,y2)
hi_addblocklabel(x6+xi,y2+yi)
-- add a new blocklabel at x,y
hi_selectlabel(x6+xi,y2+yi)
-- select the blocklabel closest to x,y
hi_setblockprop("coilS",0,ElemSize,1) -- define the region properties
hi_clearselected()
-- clear previous selections
-- background air-region
hi_addnode(x0,y0)
-- add a new node at x,y
hi_addnode(x8,y0)
hi_addnode(x8,y5)
hi_addnode(x0,y5)
hi_addsegment(x0,y0,x8,y0)
-- add a new line segment from x1,y1 to x2,y2
hi_addsegment(x8,y0,x8,y5)
hi_addsegment(x8,y5,x0,y5)
hi_addsegment(x0,y5,x0,y0)
hi_addblocklabel(x0+xi,y0+yi)
-- add a new blocklabel at x,y
hi_selectlabel(x0+xi,y0+yi)
-- select the blocklabel closest to x,y
hi_setblockprop("air",0,ElemSize,1) -- define the region properties
hi_clearselected()
-- clear previous selections
-- specify boundary conditions for the bounding air-region
4. Numerical field modelling of a transformer
33
hi_selectsegment(0.5*(x1+x7),y4) -- select the line closest to x,y
hi_selectsegment(x7,0.5*(y2+y3))
hi_selectsegment(x1,0.5*(y2+y3))
hi_setsegmentprop("Tamb",0,0)
-- set the line segment properties
hi_clearselected()
-- clear previous selections
hi_zoomnatural()
-- view all geometry
Tmin=Tcoil-DeltaT
Tmax=Tcoil
hi_saveas("tmp_heat_ini.feh")
hi_createmesh()
hi_analyse()
hi_loadsolution()
-----
save the file
create a mesh
solve the problem
load the solution
-- iterative computation of max current density
weight=0.025;
-- weight of iteration
nriter=10;
-- max number of iterations
err=1; hh=0;
-- initial conditions
h=openfile("tmp_heat_res.txt","a"); -- open an output file for writing
while err==1 do
-- iterative loop
hh=hh+1; Tmax=0;
if hh==1 then jc2=Jcm; else jc2=jc1; end
ho_seteditmode("point")
for kk = 0,10,1 do
-- find maximum temperature in the line
xp=x3+(x4-x3)/10*kk;
T,Fx,Fy,Gx,Gy,kx,ky= ho_getpointvalues(xp,0.5*(y2+y3));
if T>Tmax then Tmax=T; xpmax=xp; end
end
-- give a new value to current density
if Tcoil-Tmax < -40 then jc1=jc2*0.5;
elseif Tcoil-Tmax > 40 then jc1=jc2*2;
else
jc1=jc2+weight*(Tcoil-Tmax);
end
ho_seteditmode("area")
ho_selectblock(0.5*(x3+x4),0.5*(y2+y3));
Tave=ho_blockintegral(0);
-- find an average temperature
ho_clearblock()
ho_close()
rcu=rcu0*(1+0.004*(Tave-20));
pcu=rcu*(jc1*1e+6)^2*Kf;
-- give a new value for specific loss
hi_modifymaterial("coilP",3,pcu)
hi_modifymaterial("coilS",3,pcu)
hi_saveas("tmp_heat_ini.feh")
hi_analyse()
-- analyse updated problem
hi_loadsolution()
ho_showdensityplot(1,1,0,Tmax,Tmin) -- show the temperature plot
Terr=abs(Tmax-Tcoil);
-- conditions to finish the iterative loop
if Terr<0.01 then err=0; end
if hh>nriter then err=0; end
-- messagebox(jc1)
end
for kk = 0,10,1 do
-- find maximum temperature in the middle line
xp=x3+(x4-x3)/10*kk;
T,Fx,Fy,Gx,Gy,kx,ky= ho_getpointvalues(xp,0.5*(y2+y3));
if T>Tmax then Tmax=T; xpmax=xp; end
end
ho_selectblock(0.5*(x3+x4),0.5*(y2+y3));
Tave=ho_blockintegral(0);
-- find an average temperature
Aw1 =ho_blockintegral(1);
-- find coil cross-section area
ho_clearblock()
4. Numerical field modelling of a transformer
34
write(h,xpmax," ",Tmax," ",Tave," ",pcu," ",jc1," ",Aw1,"\n")
closefile(h)
-- close the file for writing
ho_savebitmap(filename)
-- saves the current view to specified file
Q.14 Is the core losses temperature dependent?
Q.15 How suppose to be distributed coils that they could have equal thermal loading?
4.5 Computation sequence
The main task in the parameterised command routine is to carry out a series of computation and
obtain numeric values into the table below that investigates the optimal proportions of the
transformer. The numeric values from column 2 to 6 are directly obtained from FEMM and
Mirage. The valued in column 7 is the product of columns 2 and 4 times copper fill-factor. The
apparent power is estimated according to equation 3.18 and the losses are estimated from the loss
density and the volume of the circuit.
Table 4.1 the result of the computation sequence of a transformer
Flux linkage for the
primary winding
Ampere turns for the
primary winding
Apparent power
according to
temperature rise
ϑcoilave
°C
97.3
104.6
106.1
106.3
106.6
107.2
108.0
109.4
112.7
Ψ1
mVs
1.076
0.958
0.841
0.721
0.600
0.480
0.360
0.240
0.120
NI1
Aturns
96.7
207.5
314.8
419.0
520.4
619.1
709.5
785.1
822.8
S1
VA
16.41
31.29
41.54
47.39
49.05
46.68
40.12
29.60
15.50
Core losses
Coil average
temperature
Aw1
Jmag
Jmax
2
2
mm
A/mm A/mm2
8.0
8.08
20.16
32.0
2.52
10.80
72.0
1.45
7.28
128.0
0.94
5.45
200.0
0.70
4.33
288.0
0.54
3.58
392.0
0.46
3.01
512.0
0.38
2.55
648.0
0.37
2.11
Copper losses
Current density
according to maximum
temperature
Current density
according to initial
magnetization
2
4
6
8
10
12
14
16
18
Coil cross-section area
Coil width
ww
mm
Pcu
W
5.87
6.91
7.11
7.09
7.00
6.89
6.66
6.28
5.50
Pfe
W
2.25
2.18
2.06
1.91
1.70
1.45
1.15
0.81
0.43
5 A three-phase transformer
This chapter gives a short overview of three-phase transformers, their construction and
connections [6][26].
5.1 Construction
Similar to the constructions of a single-phase transformer, the three-phase transformer with a
common core can be a shell-type (Figure 5.1), where the core surrounds the coil or a core-type
(Figure 5.2), where the coil surrounds the core. The three-phase transformer with a common core is
smaller than a bank of three single-phase transformers of the same total rating. This is due to
symmetric loading of magnetic fluxes the magnetic core. One of the simplest core arrangements is
to use a pair of E shell laminations to form a three phase core, which has also a drawback of poorly
utilised ‘slot’ areas in the sides. Independent of a staked lamination or a wound core the three phase
winding is always mounted and connected symmetrically.
Figure 5.1 A shell type of a single-phase transformer and a three-phase transformer.
Figure 5.2 A core type of a single-phase transformer and a three-phase transformer.
5. A three-phase transformer
36
The example calculation of the single-phase and a corresponding three-phase transformer show the
magnetic flux path ad flux density when the magnetizing current (in phase with flux) is maximum at
phase-a and has the corresponding current density about 0.2 A/mm2. The magnetic core is made of
electromagnetic steel M530-50A and has the same width and height. The length of the core is made
proportionally longer for the three phase counterpart. The slot area of a single phase in the threephase transformer is an half of the available area in the corresponding single-phase transformer. At
the same time the neighbouring phases contribute the magnetizing flux in the transformer. This
example makes the single-phase transformer comparable with the three-phase transformer and
visualizes the proportional relation as well as the geometry as the magnetization.
Q.16 How should be transformer constructed in order to handle an unbalanced loading?
Q.17 How circulating currents influence transformer’s performance?
5.2 Electric circuits
The windings of the three phase transformer can be connected either to star or wye (Y) connection,
delta (Δ) connection, or open star or delta connection. Figure 5.3 shows the schematic of the threephase transformer where the primary is delta connected to the supply and the secondary is star
(wye) connected to the load.
A
a
B
b
C
c
Figure 5.3 a schematic of a delta/wye transformer
•
Y/Y connection enables the primary and secondary voltages to be in phase or inverted i.e.
having 1800 a phase shift. The neutral connections are optional, and their inclusion or
omission gives different characteristics for the transformer. Sensitive to voltage unbalance.
•
Y/Δ connection causes always 300 phase shift that can be either leading or lagging between
the secondary line-to-line voltages to the nearest primary line-to-line voltage. The neutral
connections are optional, and their inclusion or omission gives different characteristics for
the transformer. Less dependent on voltage unbalance and can operate efficiently with
asymmetric supply. Possible connection is also open wye/open delta.
•
Δ/Y connection causes 300 phase shift and has similar characteristics like the previous
connection.
•
Δ/ Δ connection causes always 300 phase shift that can be either leading or lagging
between the secondary line-to-line voltages to the nearest primary line-to-line voltage. The
neutral connections are optional, and their inclusion or omission gives different
characteristics for the transformer. Possible connection is also open delta/open delta.
6 Experimental work
Prototyping transformer originates a great deal of practical questions that have been usually
neglected or not exactly considered in the design process. The questions like mechanical tolerances
margins, inter-winding and interlayer insulation, material and assembling properties. The design
procedure can take into consideration of the manufacturing procedure and not only the tables of
standard lamination shapes and the cross-sections of an enamelled wire. To understand the
terminology is as important as to understand the technology.
6.1 Prototyping
Typically, when constructing a transformer the core is selected according to the application
requirements, which is the outcome of the design process. This is followed by specifying the layout
of the windings where the primary turns and the secondary turns are computed. The winding
together with insulation is to be fitted into core window. The winding is usually wound on the
bobbin and the stack of the lamination placed into the coil window. If everything fits together with
respect to the tolerances then it is in safe hands to complete the prototype of a transformer.
6.2 Measurements
The transformer characteristics can be obtained from the equivalent circuit parameters. The
parameters of the approximate equivalent circuit are readily obtained from open-circuit and shortcircuit tests.
•
The resistance of the winding(s) can be obtained with Ohmmeter.
•
The primary inductance can be measured from open-circuited transformer. The inductance
is a consequence of the iron and air in the magnetic field path, and is non-linear, which
means that the inductance has different values under different conditions.
•
Leakage inductance is inductance that results from the parts of the primary's magnetic field
that does not link the secondary. This is an inductance from which the secondary can never
draw energy, and represents a loss of effectiveness in the transformer. This can be obtained
from the short-circuited transformer.
•
The core losses can be found from the open-circuited transformer measurements by
measuring the voltage across the winding U0, the magnetizing current I0 and the input
active power P0.
Rc =
U 02
P0
( 6.1 )
Y0 =
I0
U0
( 6.2 )
Xm =
•
1
Y02 − 1 / Rc2
( 6.3 )
By a similar experiment the leakage impedance can be found from the short-circuited
transformer. As the magnetizing impedance differs two orders of magnitude from the
leakage impedance, the parallel coupled magnetizing circuit can be ignored.
6. Experimental work
38
Re =
Pss
I ss2
( 6.4 )
Ze =
U ss
I ss
( 6.5 )
X e = Z e2 − Re2
( 6.6 )
These previously described measurements base on the simplified equivalent circuit that is shown in
Figure 6.1. In order to keep the test voltage and current lower, the open-circuit test can be carried
out from the secondary side and the short-circuit test from the primary side.
Figure 6.1 Simplified equivalent circuit
•
The thermal loading can be estimated by winding resistance or thermocouple
measurements.
•
The efficiency is determined from the measurement of losses.
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